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Picture fuzzy set is an extension of intuitionistic fuzzy set, which allows interpreting uncertain data in decisionmaking problems. This study provides aggregation operators based on Choquet integral, namely Choquet integral picture fuzzy geometric aggregation (CIPFGA) operator and Choquet integral picture fuzzy hybrid geometric aggregation (CIPFHGA) operator with certain properties of these operators are established. We validate the functioning of the operators with illustrative examples. The CIPFHGA operator has an added benefit of combining the weights of positions along with capturing the comprehensive correlative relationships of the criteria. Further, the proposed operators enable us to solve a numerical problem in picture fuzzy Multi attribute decisionmaking problem and also allows to make a comparison study with the existing literature.
aggregation operators, Choquet integral, fuzzy measure, multi attribute decisionmaking problem, picture fuzzy set
Multi attribute decisionmaking (MADM) is the process of selecting the most suitable alternative from among a number of available alternatives. Zadeh [1] introduced the main concept of fuzzy set (FS) theory in 1965 and examined the degree of membership of an object in a set, which ranges from zero to one. Later, Intuitionistic fuzzy set (IFS) was established by Atanassov [2] in 1986 as an extension of FS by including the degree of nonmembership as well as the degree of membership to each object in a set and demonstrating various properties associated with operations and relations over sets. If an expert obtains an opinion from a certain person regarding any object, that person may indicate that 0.5 is the likelihood that the statement is true, 0.3 is the possibility that the statement is false and 0.2 is that he or she is not sure. As a result, the role of neutrality is not handled by FSs or IFSs but must be addressed in reallife decision making. Cuong [3, 4] proposed a picture fuzzy set (PFS) and investigated certain essential operations and properties. The PFS is influenced by the degree of membership, the degree of neutral membership and the degree of nonmembership. The only requirement is that the sum of the three degrees be less than or equal to one. Essentially, PFS based models may be applied to situations requiring human opinions involving more sorts of answers: yes, abstain, no, refuse, which cannot be adequately described in FS and IFS. Dutta et al. [5] defined (α, δ, β)cuts under picture fuzzy (PF) environment. Many researchers introduced PFSs into the MADM field to express fuzzy evaluations due to their high performance in dealing fuzzy information.
Aggregation operators (AOs) are especially important in the decisionmaking (DM) process since they combine all of the provided individual assessment values into a single form. Yager [6] explored the features of an ordered weighted AOs and described a new form of AOs. Cuong et al. [7] investigated picture tnorm and picture tconorms and its properties for PFSs. Wei [8] described various PF Weighted average (WA) and Weighted geometric (WG) operators as well as PF hybrid aggregators and their applications. Wang et al. [9] studied MADM problems utilising some PF geometric operators. Garg [10] demonstrated a series of AOs for the PFSs that were used to solve the multi criteria decisionmaking problem. Wei [11] used arithmetic and geometric AOs, Hamacher operations with PF environment to study the MADM problem. Jana et al. [12] developed PF Dombi arithmetic and geometric AOs are used to solve MADM problem. Zhang et al. [13] developed new PF operational laws based on Dombi tnorm and tconorm and utilised Heronian mean (HM) information AO to combine picture fuzzy numbers (PFN) and the suggested operators not only combine individual attribute values but they are also able to model the general correlation between attributes. Wang et al. [14] used a case study regarding financial investment risk to provide various picture fuzzy aggregation operators (PFAO) that were based on the classic Muirhead mean (MM) operators and developed the MADM approach. Xu et al. [15] proposed a family of MM operators with PF environment and its properties are investigated. Jana et al. [16] developed MADM approaches for enterprise performance evaluation using PF Hamacher AOs. Khan et al. [17] suggested an AOs based on PF Einstein operations for the MADM problem using PF environment. Qiyas et al. [18] developed some AO based on the idea of yager operators with PF environment based on Archimedean tnorm and tconorm. Qin et al. [19] proposed a novel MADM approach that uses a set of Archimedean power Maclaurin symmetric mean (MSM) operators of PFNs. Tapan Senapati [20] introduced the AOs of PFNs and a few AOs mainly PF AczelAlsine average AOs as well as these operators to develop a method for solving MADM in a PF environment. Table 1 is a list of the AOs of PFNs on which these approaches are based.
Table 1. Existing aggregation operators of PFNs
Authors 
Aggregation operators of PFNs 
Operation 
Wei [8] 
WA, WG, Ordered WA, Ordered WG, Hybrid average (HA) and Hybrid geometric (HG) operators. 
Algebraic 
Garg [10] 
WA, Ordered WA and HA operators. 
Archimedean 
Wei [11] 
HamacherWA, Hamacher Ordered WA, HamacherHA, HamacherWG, Hamacher Ordered WG, HamacherHG, Hamacher Correlated averaging (CA) and Hamacher Correlated geometric (CG), Induced Hamacher Ordered WA, Induced Hamacher: Ordered WG, Induced Hamacher CA and Induced Hamacher CG, Hamacher weighted prioritised average, prioritised geometric, power average and power geometric operators. 
Hamacher 
Jana et al. [12] 
Dombi WA, Ordered WA, HA, WG, Ordered WG, HG operators. 
Dombi 
Zhang et al. [13] 
Dombi weighted Heronian mean, Dombi weighted dual Heronian mean operators. 
Dombi 
Xu et al. [15] 
Weighted Muirhead mean, Weighted dual Muirhead mean operators. 
Algebraic 
Jana et al. [16] 
Hamacher WA and WG operators. 
Hamacher 
khan et al. [17] 
Einstein WA and Ordered WA operators. 
Einstein 
Qiyas et al. [18] 
Yager WA, Ordered WA, Hybrid WA, WG, Ordered WG, Hybrid WG operators. 
Yager 
Qin et al. [19] 
Archimedean power MSM and Archimedean power weighted MSM operators. 
Archimedean 
Senapati [20] 
Aczel–Alsina WA, Ordered WA and HA operators. 
Aczel–Alsina 
The above mentioned PFAOs are initiated within a presumption: The criteria are independent. In most of the MADM problems which are dependent on the contrary, they are correlative. In such cases, the important degrees of criteria are given as fuzzy measures (FM) rather than weights, because weighted AOs cannot be aggregated. FMs can well reflect the correlative relationships between criteria sets, such as redundant, independent, and complimentary. In reality, FMs are extensions of weights due to the fact that the sum of all FMs of criteria set can exceed one, whereas the FM of the entire set_is_restricted to one. Consequently, the FMs are more flexible. Choquet integral is an efficient method for aggregating ranking based on correlated criteria. To handle multicriteria group decision making (MCGDM) problems where the criteria and expert opinions frequently involve interdependent or interaction phenomena among criteria. Tan et al. [21] investigated the generalised IF ordered geometric averaging operator. Zhang et al. [22] developed the Einstein interval IF Choquet geometric operator and investigated the relationship with the IF Choquet geometric operator. Singh et al. [23] described Choquet averaging and geometric mean operators for PFSs and also presented a VIKOR for MCGDM problems based on PF Choquet integrals. IF arithmetic and hybrid arithmetic AOs are proposed by Jia and Wang [24] based on Choquet integral as well as proposed operator computes the correlative criteria and combines the weight of position.
In MADM problems, the decisionmaking criteria may be dependent or independent. FMs were first presented by Sugeno [25] in 1974 to model interactions between decision criteria. The aforementioned AOs, which are unable to properly handle the specific conditions with correlative criteria and weighted positions, we define an AOs as the Choquet integral picture fuzzy geometric aggregation (CIPFGA) operator. After that, we combine the FMs of the criteria and the weights of the positions to construct the Choquet integral picture fuzzy hybrid geometric aggregation (CIPFHGA) operator.
The summary of the motivations of this article are outlined as follows:
The summary of the contributions of this article are outlined as follows:
Table 2. A comparison of the existing operators and the proposed operator

Weight of the criteria 
Fuzzy measure and weight of the criteria 
Correlative relationships among criterion sets 
Wei [8] 
$\sqrt{\quad}$ 
$\times$ 
$\times$ 
Wang et al. [9] 
$\sqrt{\quad}$ 
$\times$ 
$\times$ 
Garg [10] 
$\sqrt{\quad}$ 
$\times$ 
$\times$ 
Jana et al. [16] 
$\sqrt{\quad}$ 
$\times$ 
$\times$ 
Khan et al. [17] 
$\sqrt{\quad}$ 
$\times$ 
$\times$ 
Qiyas et al. [18] 
$\sqrt{\quad}$ 
$\times$ 
$\times$ 
Proposed operator 
$\sqrt{\quad}$ 
$\sqrt{\quad}$ 
$\sqrt{\quad}$ 
From Table 2, existing AOs [810, 1618] are not dealing with FMs and correlative relationships among criteria sets, whereas the proposed CIPFHGA operator completely deals with them.
The present article is structured in a comprehensive manner into seven detailed sections: In Section 2, we briefly review the concepts with respect to the PFSs and the fuzzy measures. In Section 3, the CIPFGA and CIPFHGA operators are defined in detail, along with their properties. Section 4, a MADM approach that relies on the CIPFHGA operator is proposed. Section 5, applying the proposed MADM approach to the numerical problem to demonstrate its practicality. Section 6, Comparative analyses are conducted to demonstrate the advantages of our suggested MADM approach over existing aggregation operators. Conclusions and potential study directions are discussed in the final section.
This section provides the definitions required to support the proposed research work.
2.1 Picture fuzzy set
Cuong [3, 4] developed the PFS as an extension of the IFS. PFS is represented mathematically as follows:
Definition 1: A picture Fuzzy “sets A on universal set X is defined by:
$\mathrm{A}=\left\{\left\langle x, \mu_{\mathrm{A}}(x), \eta_{\mathrm{A}}(x), v_{\mathrm{A}}(x)\right\rangle \mid x \in X\right\}$
where, $\mu_A(x), \eta_A(x)$ and $v_A(x) \in[0,1]$ are the degree of membership, the degree of neutral membership and the degree of nonmembership of $x \in A$ respectively, with the following condition: $0 \leq \mu_A(x)+\eta_A(x)+v_A(x) \leq 1 \quad \forall x \in X$. Then for $x \in X$, $\pi_A(x)=1\mu_A(x)\eta_A(x)v_A(x)$ could be called the degree of refusal membership of $x$ in $A$. For convenience, let $\alpha=\left(\mu_\alpha, \eta_\alpha, v_\alpha\right)$ is called a PFN.
2.2 Comparison for picture fuzzy numbers
According to Garg [10] the score and accuracy functions are as follows:
Definition 2: Let α=(μ_{α}, η_{α}, ν_{α}) be a PFN and its score function S(α) and its accuracy function A(α) is defined by:
$\begin{array}{cl}S(\alpha)=\mu_\alpha\eta_\alphav_\alpha ; & S(\alpha) \in[1,1], \\ A(\alpha)=\mu_\alpha+\eta_\alpha+v_\alpha ; & A(\alpha) \in[0,1]\end{array}$.
Based on the S(α) and A(α)an order relationship between two PFNs is defined as follows.
Definition 3: Let $\alpha_1=\left(\mu_{\alpha_1}, \eta_{\alpha_1}, \nu_{\alpha_1}\right)$ and $\alpha_2=\left(\mu_{\alpha_2}, \eta_{\alpha_2}, v_{\alpha_2}\right)$ are two PFNs, the following is their order using the score and accuracy functions:
(1) If S(α_{1}) > S(α_{2}) then α_{1 }> α_{2};
(2) If S(α_{1}) = S(α_{2}) then
(a) If A(α_{1}) > A(α_{2}) then α_{1 }> α_{2};
(b) If A(α_{1}) = A(α_{2}) then α_{1 }= α_{2}.
2.3 Operation laws of picture fuzzy numbers
Wang et al. [9] following operational laws are defined for PFNs.
Definition 4: Le $\alpha_1=\left(\mu_{\alpha_1}, \eta_{\alpha_1}, v_{\alpha_1}\right)$ and $\alpha_2=$ $\left(\mu_{\alpha_2}, \eta_{\alpha_2}, v_{\alpha_2}\right)$ are two PFNs and $\lambda>0$. Then PFN operations can be defined as follows:
$~{{\alpha }_{1}}\otimes {{\alpha }_{2}}=\left\langle \begin{align} & \left( {{\mu }_{{{\alpha }_{1}}}}+{{\eta }_{{{\alpha }_{1}}}} \right)\left( {{\mu }_{{{\alpha }_{2}}}}+{{\eta }_{{{\alpha }_{2}}}} \right){{\eta }_{{{\alpha }_{1}}}}{{\eta }_{{{\alpha }_{2}}}},{{\eta }_{{{\alpha }_{1}}}}{{\eta }_{{{\alpha }_{2}}}}, \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\left( 1{{\nu }_{{{\alpha }_{1}}}} \right)\left( 1{{\nu }_{{{\alpha }_{2}}}} \right) \\\end{align} \right\rangle $
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\alpha }_{1}}^{\lambda }=\left\langle {{\left( {{\mu }_{{{\alpha }_{1}}}}+{{\eta }_{{{\alpha }_{1}}}} \right)}^{\lambda }}\eta _{{{\alpha }_{1}}}^{\lambda },\eta _{{{\alpha }_{1}}}^{\lambda },1{{\left( 1{{\nu }_{{{\alpha }_{1}}}} \right)}^{\lambda }} \right\rangle .$
These two operational laws are also satisfying the following properties under condition λ, λ_{1}, λ_{2 }> 0.
(1) $\alpha_1 \otimes \alpha_2=\alpha_2 \otimes \alpha_1$
(2) $\left(\alpha_1 \otimes \alpha_2\right)^\lambda=\alpha_1^\lambda \otimes \alpha_2^\lambda$
(3) $\alpha_1^{\lambda_1} \otimes \alpha_1^{\lambda_2}=\alpha_1^{\lambda_1+\lambda_2}$.
2.4 Fuzzy measure and existing PFCGM operator
Analyzing the relationships between a set of criteria is accomplished using FMs. Unlike weights, which only represent the importance of a single criteria, FMs properly reflect the importance of a collection of criteria. Fuzzy measure Sugeno [25] and picture fuzzy Choquet geometric mean (PFCGM) operator Singh et al. [23] are discussed in this section.
Definition 5: Sugeno et al. [25] Let S={S_{1}, S_{2}, ..., S_{n}} be an universe of discourse, a set function ϑ: P(S) → [0, 1] is a FM on S, if satisfying the following conditions:
(1) $\vartheta(\phi)=0, \vartheta(S)=1$；
(2) If $R_1, R_2 \subseteq P(S)$ and $R_1 \subseteq R_2$, then $\vartheta\left(R_1\right) \leq \vartheta\left(R_2\right)$.
Sugeno [25] suggested a λfuzzy measure, which may be described as follows:
$\vartheta \left( {{R}_{1}}\cup {{R}_{2}} \right)=\vartheta \left( {{R}_{1}} \right)+\vartheta \left( {{R}_{2}} \right)+\lambda \cdot \vartheta \left( {{R}_{1}} \right)\vartheta \left( {{R}_{2}} \right)$
$\forall {{R}_{1}},{{R}_{2}}\subseteq P\left( S \right)$ and ${{R}_{1}}\cap {{R}_{2}}=\phi ,$ $\lambda \ge 1$
λ diagnoses the relationship between the criteria in this case. For different values of λ, the λ FM reduces to various measures. If λ = 0, then it becomes simply additive measure, if λ >0 then it becomes to superadditive measure, and if λ < 0 then it becomes to sub additive measure.
$\vartheta \left( S \right)=\left( \begin{matrix} \frac{1}{\lambda }\left( \prod\limits_{i=1}^{n}{\left( 1+\lambda \cdot \vartheta \left( {{S}_{i}} \right) \right)1} \right) & if\,\lambda \ne 0 \\ \sum\limits_{i=1}^{n}{\vartheta \left( {{S}_{i}} \right)} & if\,\lambda =0 \\\end{matrix} \right)$ (1)
Eq. (1), boundary condition ϑ(S_{i}) = 1 can be used to find parameter λ, following Eq. (2) is similar to solve Eq. (1).
$\lambda +1=\left( \prod\limits_{i=1}^{n}{\left( 1+\lambda \vartheta \left( {{S}_{i}} \right) \right)} \right)$ (2)
Definition 6: Singh et al. [23] Let$\alpha_j=\left(\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right)(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and fuzzy measure on S. The PFCGM operator of α_{i} with respects to ϑ is defined as follows:
$PFCGM\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)\,=\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{\left( {{\mu }_{{{\alpha }_{\left( j \right)}}}} \right)}^{\vartheta \left( {{R}_{\left( j \right)}} \right)\vartheta \left( {{R}_{\left( j+1 \right)}} \right)}},1\prod\limits_{j=1}^{n}{{{\left( 1{{\eta }_{{{\alpha }_{\left( j \right)}}}} \right)}^{\vartheta \left( {{R}_{\left( j \right)}} \right)\vartheta \left( {{R}_{\left( j+1 \right)}} \right)}}},} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\prod\limits_{j=1}^{n}{{{\left( 1{{\nu }_{{{\alpha }_{\left( j \right)}}}} \right)}^{\vartheta \left( {{R}_{\left( j \right)}} \right)\vartheta \left( {{R}_{\left( j+1 \right)}} \right)}}} \\\end{align} \right\rangle$
where, (j) denotes a permutation on C such that α_{(j) }≤ α_{(j+1)}, j = 1, 2, ⋯, n1 and R_{(j) }= {S_{(j)}, S_{(j+1)}, ⋯, S_{(n)}} with R_{(n+1) }= ϕ.
The Choquet integral based CIPFGA operator is defined in this section and it is combined with the FMs of criteria and the weights of positions to define the CIPFHGA operator.
3.1 CIPFGA operator
Wei [8] and Garg [10] presented certain AOs to aggregate PF environment for MADM problems under the premise of independent criteria. In this part, we propose the Choquet integral operator for the PF environment, which provides for the interaction phenomena between the criteria represented by FM. The CIPFGA operator is defined as follows.
Definition 7: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and FM on S. The discrete choquet integral CIPFGA operator of α_{j} with respect to ϑ is define as
$\begin{align} & CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,={{\alpha }_{1}}\frac{\sum\limits_{{{R}_{1}}\subseteq {{P}_{1}}\left( C \right)}{\vartheta \left( {{R}_{1}}\cup \left\{ S \right\} \right)\vartheta \left( {{R}_{1}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}}\oplus \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\alpha }_{2}}\frac{\sum\limits_{{{R}_{2}}\subseteq {{P}_{2}}\left( C \right)}{\vartheta \left( {{R}_{2}}\cup \left\{ {{S}_{2}} \right\} \right)\vartheta \left( {{R}_{2}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}}\oplus \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\cdots \oplus {{\alpha }_{n}}\frac{\sum\limits_{{{R}_{n}}\subseteq {{P}_{n}}\left( C \right)}{\vartheta \left( {{R}_{n}}\cup \left\{ {{S}_{n}} \right\} \right)\vartheta \left( {{R}_{n}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}} \\\end{align}$ (3)
where, $R_j \subseteq P_j(S), P_j(S)=P\left(S\left\{S_j\right\}\right)$ Indicates the power of S. For convenience, let ${{\beta }_{j}}=\frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}},$ then Eq. (3) can be simplified as $CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\underset{j=1}{\overset{n}{\mathop{\otimes }}}\,{{\alpha }_{j}}^{{{\beta }_{j}}}.$
Theorem 1: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and FM on S. Using the CIPFGA operator, their aggregated value is also a PFN and
$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},\cdots ,{{\alpha }_{n}} \right)\,=\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$ (4)
where, ${{\beta }_{j}}=\frac{\sum\limits_{{{B}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}},$ $R_j \subseteq P_j(S), P_j(S)=P\left(S\left\{S_i\right\}\right)$.
Proof
From the definition (4), the first results come quickly Below, we use mathematical induction on n to show that Eq. (4).
When n=2, since:
$\alpha _{1}^{{{\beta }_{1}}}=\left\langle {{\left( {{\mu }_{{{\alpha }_{1}}}}+{{\eta }_{{{\alpha }_{1}}}} \right)}^{{{\beta }_{1}}}}\eta _{{{\alpha }_{1}}}^{{{\beta }_{1}}},\eta _{{{\alpha }_{1}}}^{{{\beta }_{1}}},1{{\left( 1{{\nu }_{{{\alpha }_{1}}}} \right)}^{{{\beta }_{1}}}} \right\rangle $
$\alpha _{2}^{{{\beta }_{2}}}=\left\langle {{\left( {{\mu }_{{{\alpha }_{2}}}}+{{\eta }_{{{\alpha }_{2}}}} \right)}^{{{\beta }_{2}}}}\eta _{{{\alpha }_{2}}}^{{{\beta }_{2}}},\eta _{{{\alpha }_{2}}}^{{{\beta }_{2}}},1{{\left( 1{{\nu }_{{{\alpha }_{2}}}} \right)}^{{{\beta }_{2}}}} \right\rangle $
Then, $\operatorname{CIPFGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=\alpha_1^{\beta_1} \otimes \alpha_2^{\beta_2}=$$\left\langle\begin{array}{l}\left(\mu_{\alpha_1}+\eta_{\alpha_1}\right)^{\beta_1}\left(\mu_{\alpha_2}+\eta_{\alpha_2}\right)^{\beta_2}\eta_{\alpha_1}^{\beta_1} \eta_{\alpha_2}^{\beta_2}, \\ \eta_{\alpha_1}^{\beta_1} \eta_{\alpha_2}^{\beta_2}, 1\left(1v_{\alpha_1}\right)^{\beta_1}\left(1v_{\alpha_2}\right)^{\beta_2}\end{array}\right\rangle$.
If Eq. (4) holds for n=k, that is:
$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},\cdots ,{{\alpha }_{k}} \right)\,=\left\langle \begin{align} & \prod\limits_{j=1}^{k}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{k}{{{(1{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$
Then, for n=k+1, we have:
$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{k+1}} \right)=\left\langle \begin{align} & \prod\limits_{j=1}^{k}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{k}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{k}{{{(1{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle \centerdot$
$\,\left\langle \begin{align} & {{({{\mu }_{{{\alpha }_{k+1}}}}+{{\eta }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}}{{({{\eta }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}}, \\ & {{({{\eta }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}},1{{(1{{\nu }_{{{\alpha }_{k+1}}}})}^{{{\beta }_{k+1}}}} \\\end{align} \right\rangle$
$\,=\left\langle \begin{align} & \prod\limits_{j=1}^{k+1}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{k+1}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{k+1}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{k+1}{{{(1{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$
That is to say, the condition n=k+1 is satisfied by Eq. (4).
As a result, Eq. (4) obtains for all n, concluding the proof of Theorem 1.
Theorem 2: (Idempotency) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1$, $2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$. If all $\alpha_j$ are equal, i.e., $\alpha_j=\alpha: \forall \mathrm{j}$, then CIPFGA $\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=\alpha$.
Proof
Since α_{j }= α = ⟨μ_{α}, η_{α}, ν_{α}⟩ according to theorem 1, we have:
$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\,\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle =\left\langle {{\mu }_{\alpha }},{{\eta }_{\alpha }},{{\nu }_{\alpha }} \right\rangle =\alpha$
where, ${{\beta }_{j}}=\frac{\sum\limits_{{{B}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}},$ $R_j \subseteq P_j(S), P_j(S)=P\left(S\left\{S_j\right\}\right)$.
Theorem 3: (Monotonicity) Let $\alpha_{1 j}=\left(\mu_{\alpha_{1 j}}, \eta_{\alpha_{1 j}}, v_{\alpha_{1 j}}\right)$ and $\alpha_{2 j}=\left(\mu_{\alpha_{2 j}}, \eta_{\alpha_{2 j}}, v_{\alpha_{2 j}}\right)(j=1,2, \ldots, n)$ be two family of PFNs on $S$ and $\vartheta$ be a FM on S. If $\mu_{\alpha_{1 j}} \leq \mu_{\alpha_{2 j}}, \eta_{\alpha_{1 j}} \geq \eta_{\alpha_{2 j}}$ and $v_{\alpha_{1 j}} \geq v_{\alpha_{2 j}^{\prime}} \quad$ then $\operatorname{CIPFAA}\left(\alpha_{11}, \alpha_{12}, \ldots, \alpha_{1 n}\right) \leq$ $\operatorname{CIPFAA}\left(\alpha_{21}, \alpha_{22}, \ldots, \alpha_{2 n}\right)$.
Proof
$\,CIPFGA\left( {{\alpha }_{11}},{{\alpha }_{12}},...,{{\alpha }_{1n}} \right)=\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{1j}}}}+{{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$
$CIPFGA\left( {{\alpha }_{21}},{{\alpha }_{22}},...,{{\alpha }_{2n}} \right)\,=\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{2j}}}}+{{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle $
Since $\mu_{\alpha_{1 j}} \leq \mu_{\alpha_{2 j}}, \eta_{\alpha_{1 j}} \geq \eta_{\alpha_{2 j}}$ and $v_{\alpha_{1 j}} \geq v_{\alpha_{2 j}}$ we have:
$\prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{1j}}}}+{{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}$$\le \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{2j}}}}+{{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}},$
$\prod\limits_{j=1}^{n}{{{\left( {{\eta }_{{{\alpha }_{1j}}}} \right)}^{{{\beta }_{j}}}}}\ge \prod\limits_{j=1}^{n}{{{\left( {{\eta }_{{{\alpha }_{2j}}}} \right)}^{{{\beta }_{j}}}}}$ and $1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}\ge 1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}.$
Hence
$\prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{1j}}}}+{{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{1j}}}})}^{{{\beta }_{j}}}}}}$
$\le \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{2j}}}}+{{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}},\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{2j}}}})}^{{{\beta }_{j}}}}}}$
i.e.,
$\operatorname{CIPFGA}\left(\alpha_{11}, \alpha_{12}, \ldots, \alpha_{1 n}\right) \leq \operatorname{CIPFGA}\left(\alpha_{21}, \alpha_{22}, \ldots, \alpha_{2 n}\right)$
Theorem 4: (Boundedness) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, \nu_{\alpha_j}\right\rangle(j=1$ $2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on S.
If $\alpha^{}=\left\langle\min _j\left(\mu_{\alpha_j}\right), \max _j\left(\eta_{\alpha_j}\right), \max _j\left(v_{\alpha_j}\right)\right\rangle$ and $\alpha^{+}=$ $\left(\max _j\left(\mu_{\alpha_j}\right), \min _j\left(\eta_{\alpha_j}\right), \min _j\left(v_{\alpha_j}\right)\right\rangle \quad$ then $\quad \alpha^{} \leq$ $\operatorname{CIPFGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right) \leq \alpha^{+}$.
Proof
Using theorem 2, α^{} and α^{+ }can be defined as CIPFGA(α^{}, α^{}, ..., α^{}) and CIPFGA (α^{+}, α^{+}, ..., α^{+}) respectively.
Since
${{\alpha }^{}}=\left\langle \underset{j}{\mathop{\min }}\,\left( {{\mu }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\max }}\,\left( {{\eta }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\max }}\,\left( {{\nu }_{{{\alpha }_{j}}}} \right) \right\rangle =\left\langle {{\mu }_{{{\alpha }^{}}}},{{\eta }_{{{\alpha }^{}}}},{{\nu }_{{{\alpha }^{}}}} \right\rangle$
${{\alpha }^{+}}=\left\langle \underset{j}{\mathop{\max }}\,\left( {{\mu }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\min }}\,\left( {{\eta }_{{{\alpha }_{j}}}} \right),\underset{j}{\mathop{\min }}\,\left( {{\nu }_{{{\alpha }_{j}}}} \right) \right\rangle =\left\langle {{\mu }_{{{\alpha }^{+}}}},{{\eta }_{{{\alpha }^{+}}}},{{\nu }_{{{\alpha }^{+}}}} \right\rangle$
Then $\mu_{\alpha^{}} \leq \mu_{\alpha_j}, \eta_{\alpha^{}} \geq \eta_{\alpha_j}, v_{\alpha^{}} \geq v_{\alpha_j}, \mu_{\alpha^{+}} \geq \mu_{\alpha_j}, \eta_{\alpha^{+}} \leq$ $\eta_{\alpha_j}$ and $\nu_{\alpha^{+}} \leq v_{\alpha_j} \forall j$. According to Theorem 3, we have:
$\begin{align} & CIPFGA\left( {{\alpha }^{}},{{\alpha }^{}},...,{{\alpha }^{}} \right)\le CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right) \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\le CIPFGA\left( {{\alpha }^{+}},{{\alpha }^{+}},...,{{\alpha }^{+}} \right). \\\end{align}$
Theorem 5: (Permutation) Let $\alpha_j=\left(\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right)$ and $\alpha_{(j)}=\left(\mu_{\alpha_{(j)}}, \eta_{\alpha_{(j)}}, v_{\alpha_{(j)}}\right)(j=1,2, \ldots, n)$ be two family of PFNs on $S$ and $\vartheta$ be a FM on $S$, where $(j)$ indicates an arbitrary permutation of $j$. then $\operatorname{CIPFGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=$ $\operatorname{CIPFGA}\left(\alpha_{(1)}, \alpha_{(2)}, \ldots, \alpha_{(n)}\right)$.
Proof
$CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{\alpha }_{j}}}}+{{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{\alpha }_{j}}}})}^{{{\beta }_{j}}}}}} \\\end{align} \right\rangle$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left\langle \begin{align} & \prod\limits_{\left( j \right)=1}^{n}{{{({{\mu }_{{{\alpha }_{\left( j \right)}}}}+{{\eta }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}}}\prod\limits_{\left( j \right)=1}^{n}{{{({{\eta }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}}}, \\ & \prod\limits_{\left( j \right)=1}^{n}{{{({{\eta }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}},1\prod\limits_{\left( j \right)=1}^{n}{{{(1{{\nu }_{{{\alpha }_{\left( j \right)}}}})}^{{{\beta }_{\left( j \right)}}}}}} \\\end{align} \right\rangle$
$=CIPFGA\left( {{\alpha }_{\left( 1 \right)}},{{\alpha }_{\left( 2 \right)}},...,{{\alpha }_{\left( n \right)}} \right).$
The proof of Theorem 5 is complete.
Example 1: Let α_{1 }= ⟨0.4, 0.2, 0.3⟩, α_{2 }= ⟨0.3, 0.1, 0.5⟩, α_{3 }= ⟨0.6, 0.2, 0.1⟩ be three PFNs on S, S={S_{1}, S_{2}, S_{3}} and ϑ be a fuzzy measure on S. ϑ(S_{1}) = 0.2, ϑ(S_{2}) = 0.3, ϑ(S_{3}) = 0.4, ϑ(S_{1}, S_{2}) = 0.5, ϑ(S_{1}, S_{3}) = 0.6, ϑ(S_{2}, S_{3}) = 0.7, ϑ(S_{1}, S_{2}, S_{3}) = 1. Then:
$\begin{align} & CIPFGA\left( {{\alpha }_{1}},{{\alpha }_{2}},{{\alpha }_{3}} \right) \\ & \,\,\,\,\,\,\,=\left\langle 0.4,0.2,0.3 \right\rangle \frac{0.71}{3.09}+\left\langle 0.3,0.1,0.5 \right\rangle \frac{1.03}{3.09}+\left\langle 0.6,0.2,0.1 \right\rangle \frac{1.35}{3.09} \\\end{align}$
$=\left\langle \begin{array}{c} \left((0.4+0.2)^{0.2298}0.2^{0.2298} \cdot(0.3+0.1)^{0.3333} \\ 0.1^{0.3333} \cdot(0.6+0.2)^{0.4369}0.2^{0.4369}\right), \\ 0.2^{0.2298} \cdot 0.1^{0.3333} \cdot 0.2^{0.4369}, \\ 1\left((10.3)^{0.2298} \cdot(10.5)^{0.3333} \cdot(10.1)^{0.4369}\right.\end{array} \right\rangle$
$=\left\langle 0.4356,0.1587,0.3016 \right\rangle .$
3.2 CIPFHGA operator
We propose the CIPFHGA operator, which is motivated by the PFHA operator (Wang et al. [9]).
Definition 8: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and ϑ be a family of PFNs and FM on S and has an weight vector w = (w_{1}, w_{2}, ..., w_{n})^{T} with 0≤w_{j}≤1 and The (discrete) CIPFHGA operator of α_{j} with respect to w and ϑ is defined as:
$CIPFHGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\underset{j=1}{\overset{n}{\mathop{\otimes }}}\,\tilde{\alpha }_{(j)}^{{{w}_{j}}}$
where, $\tilde{\alpha}_{(j)}$ is the j^{th} largest of the $\tilde{\alpha}_j$,
${{\tilde{\alpha }}_{j}}=n\cdot {{\alpha }_{j}}\cdot \frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}},$
If FM ϑ is additive and the FMs of S_{j}(j=1, 2, ⋯, n) are equivalent, then $R_j \subseteq P_j(S)$, P_{j}(S)=P (S{S_{j}}) and n is the balancing coefficient, which functions as a balancing factor in this situation, i.e., reduces to weight $\left(\frac{1}{n}, \frac{1}{n}, \cdots, \frac{1}{n}\right)^T$ then the vector reduces to (α_{1}, α_{2},…, α_{n})^{T}.
$\left( \begin{align} & n\cdot {{\alpha }_{1}}\frac{\sum\limits_{{{R}_{1}}\subseteq {{P}_{1}}\left( C \right)}{\vartheta \left( {{R}_{1}}\cup \left\{ {{S}_{1}} \right\} \right)\vartheta \left( {{R}_{1}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}}, \\ & n\cdot {{\alpha }_{2}}\frac{\sum\limits_{{{R}_{2}}\subseteq {{P}_{2}}\left( C \right)}{\vartheta \left( {{R}_{2}}\cup \left\{ {{S}_{2}} \right\} \right)\vartheta \left( {{R}_{2}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}}, \\ & \cdots ,n\cdot {{\alpha }_{n}}\frac{\sum\limits_{{{R}_{n}}\subseteq {{P}_{n}}\left( C \right)}{\vartheta \left( {{R}_{n}}\cup \left\{ {{S}_{n}} \right\} \right)\vartheta \left( {{R}_{n}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}} \\\end{align} \right)T$
For convenience,
let ${{\beta }_{j}}=\frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}},$ it can be found that the CIPFHGA operator is three phases:
Theorem 6: Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1,2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$ and has an weight vector $w=\left(w_1\right.$, $\left.w_2, \ldots, w_n\right)^T \quad$ with $\quad 0 \leq w_j \leq 1 \quad$ and $\quad \sum_{j=1}^n w_j=1, \quad \tilde{\alpha}_{(j)}=$ $\left\langle\mu_{\widetilde{\alpha}_{(j)}}, \eta_{\widetilde{\alpha}_{(j)}}, v_{\widetilde{\alpha}_{(j)}}\right\rangle$ be the $j^{\text {th }}$ largest of $\tilde{\alpha}_j$,
${{\tilde{\alpha }}_{j}}=n\cdot {{\alpha }_{j}}\cdot \frac{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}{\sum\limits_{j=1}^{n}{\sum\limits_{{{R}_{j}}\subseteq {{P}_{j}}\left( C \right)}{\vartheta \left( {{R}_{j}}\cup \left\{ {{S}_{j}} \right\} \right)\vartheta \left( {{R}_{j}} \right)}}},$
Then, using the CIPFHGA operator, their aggregated value is also a PFN
$CIPFHGA\left( {{\alpha }_{1}},{{\alpha }_{2}},...,{{\alpha }_{n}} \right)=\left\langle \begin{align} & \prod\limits_{j=1}^{n}{{{({{\mu }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}}+{{\eta }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}}}\prod\limits_{j=1}^{n}{{{({{\eta }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}}}, \\ & \prod\limits_{j=1}^{n}{{{({{\eta }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}},1\prod\limits_{j=1}^{n}{{{(1{{\nu }_{{{{\tilde{\alpha }}}_{\left( j \right)}}}})}^{{{w}_{j}}}}}} \\\end{align} \right\rangle \,\,$
Theorem 7: (Idempotency) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=1$, $2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$ and has an weight vector $w=\left(w_1, w_2, \ldots, w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$. If all $\alpha_j(j=1,2, \ldots, n)$ are equal, i.e., $\alpha_j=$ $\alpha: \forall j$, then $\operatorname{CIPFHGA}\left(\alpha_1, \alpha_2, \cdots, \alpha_n\right)=\alpha .$
Theorem 8: (Monotonicity) Let $\alpha_{1 j}=\left(\mu_{\alpha_{1 j}}, \eta_{\alpha_{1 j}}, v_{\alpha_{1 j}}\right)$ and $\alpha_{2 j}=\left(\mu_{\alpha_{2 j}}, \eta_{\alpha_{2 j}}, v_{\alpha_{2 j}}\right)(j=1,2, \ldots, n)$ be a family of PFNs and $\vartheta$ be a FM on $S$ and has an weight vector $w=\left(w_1, w_2, \ldots\right.$, $\left.w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$. if $\mu_{\alpha_{1 j}} \leq \mu_{\alpha_{2 j}}, \eta_{\alpha_{1 j}} \geq$ $\eta_{\alpha_{2 j}}$ and $v_{\alpha_{1 j}} \geq v_{\alpha_{2 j}}$, then $\operatorname{CIPFHGA}\left(\alpha_{11}, \alpha_{12}, \cdots, \alpha_{1 n}\right) \leq$ $\operatorname{CIPFHGA}\left(\alpha_{21}, \alpha_{22}, \cdots, \alpha_{2 n}\right)$.
Theorem 9: (Boundedness) Let $\alpha_j=\left\langle\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right\rangle(j=$ $1,2, \ldots, n)$ and $\vartheta$ be a family of PFNs and FM on $S$ and has a weight" vector $w=\left(w_1, w_2, \ldots, w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$.
If $\alpha^{}=\left\langle\min _j\left(\mu_{\alpha_j}\right), \max _j\left(\eta_{\alpha_j}\right), \max _j\left(v_{\alpha_j}\right)\right\rangle$ and $\alpha^{+}=$ $\left\langle\max _j\left(\mu_{\alpha_j}\right), \min _j\left(\eta_{\alpha_j}\right), \min _j\left(v_{\alpha_j}\right)\right\rangle$, then $\alpha \leq \operatorname{CIPFHAA}\left(\alpha_1\right.$, $\left.\alpha_2, \cdots, \alpha_n\right) \leq \alpha^{+} .$.
Theorem 10: (Permutation) let $\alpha_j=\left(\mu_{\alpha_j}, \eta_{\alpha_j}, v_{\alpha_j}\right) \alpha_{(j)}=$ $\left(\mu_{\alpha_{(j)}}, \eta_{\alpha_{(j)}}, v_{\alpha_{(j)}}\right)(j=1,2, \ldots, n)$ be two familys of PFNs on $S$, $\vartheta$ be a fuzzy measure on $S$ and has an weight vector $w=\left(w_1\right.$, $\left.w_2, \ldots, w_n\right)^T$ with $0 \leq w_j \leq 1$ and $\sum_{j=1}^n w_j=1$. where $(j)$ denotes an arbitrary permutation of $j$. Then
$\operatorname{CIPFHGA}\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right)=\operatorname{CIPFHGA}\left(\alpha_{(1)}, \alpha_{(2)}, \ldots, \alpha_{(n)}\right)$.
Theorem 2  Theorem 5 proofs are similar to Theorem 7  Theorem 10 proofs, which are ignored here.
Example 2: Suppose w=(0.2429, 0.5142, 0.2429)^{T} by referring to (Xu [26]) then, the example 1, by using the CIPFHGA operator we have:
$\begin{align} & \alpha _{1}^{{{\beta }_{1}}}=0.2298\cdot \left\langle 0.4,0.2,0.3 \right\rangle =\left\langle 0.1984,0.6908,0.0787 \right\rangle \\ & \alpha _{2}^{{{\beta }_{2}}}=0.3333\cdot \left\langle 0.3,0.1,0.5 \right\rangle =\left\langle 0.2726,0.4642,0.2063 \right\rangle \\ & \alpha _{3}^{{{\beta }_{3}}}=0.4369\cdot \left\langle 0.6,0.2,0.1 \right\rangle =\left\langle 0.4121,0.4950,0.0449 \right\rangle . \\\end{align}$
Then, $\quad S\left(\alpha_1^{\beta_1}\right)=0.5711 \quad, \quad S\left(\alpha_2^{\beta_2}\right)=0.3989$ and $S\left(\alpha_3^{\beta_3}\right)=0.1278$.
Following the rules of section 2.2, we may conclude that based on these comparative laws,
${{\tilde{\alpha }}_{1}}=\alpha _{3}^{3{{\beta }_{3}}}=\left\langle 0.3734,0.3297,0.2180 \right\rangle$
${{\tilde{\alpha }}_{2}}=\alpha _{2}^{3{{\beta }_{2}}}=\left\langle 0.2999,0.1207,0.5000 \right\rangle$
${{\tilde{\alpha }}_{3}}=\alpha _{1}^{3{{\beta }_{1}}}=\left\langle 0.6251,0.1213,0.1287 \right\rangle .$
then,
$\begin{align} & CIPFHGA\left( {{\alpha }_{1}},{{\alpha }_{2}},\cdots ,{{\alpha }_{n}} \right) \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.2429\cdot \left\langle 0.3734,0.3297,0.2180 \right\rangle \otimes \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.5142\cdot \left\langle 0.2999,0.1207,0.5000 \right\rangle \otimes \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0.2429\cdot \left\langle 0.6251,0.1213,0.1287 \right\rangle \, \\\end{align}$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\left\langle 0.1556,0.1543,0.3621 \right\rangle .$
To begin, we discuss MADM problem with correlative criteria in a PF environment. Then, we provide a MADM problem that is using CIPFHGA operator.
4.1 Description of MADM problems
Let A_{i}={A_{1}, A_{2}, ⋯ A_{m}} (i=1, 2,⋯, m) is the finite set of alternative, S_{j}={S_{1}, S_{2}, ⋯ S_{n}} (j=1, 2, ⋯ n) is the finite set of criteria and the importance degree of S_{j} is denote as FM ϑ. The evaluation of the alternatives under each criteria is provide by the decision making and is denoted by the PFNs where $\mu_{\alpha_{i j}}, \eta_{\alpha_{i j}}, v_{\alpha_{i j}}$ denotes the extent to which the decisionmaking analyses whether alternative A_{i} should satisfy, neutral satisfy and non satisfy of the criteria S_{j}. The evaluation of matrix is constructed as followes.
$E=\left[ \begin{matrix} {{\alpha }_{11}} & {{\alpha }_{12}} & \cdots & {{\alpha }_{1n}} \\ {{\alpha }_{21}} & {{\alpha }_{22}} & \cdots & {{\alpha }_{2n}} \\ \vdots & \vdots & \ddots & \vdots \\ {{\alpha }_{n1}} & {{\alpha }_{n2}} & \cdots & {{\alpha }_{mn}} \\\end{matrix} \right]$
A flowchart for the proposed MADM problem is shown in Figure 1 and the detailed steps are presented as follows.
Steps of MADM problem:
Following are the steps aimed at solving the MADM problem with PF environment using the proposed operator.
Step 1: The normalized PF decision matrix is generated by defining the attribute set S_{j} into S_{1} and S_{2}, with S_{1} representing benefit attributes and S_{2} representing cost attributes. Eq. (5) is used to convert the benefit type to the cost type:
${{\alpha }_{ij}}=\left\{ \begin{matrix} \left\langle {{\mu }_{{{\alpha }_{ij}}}},{{\eta }_{{{\alpha }_{ij}}}},{{\nu }_{{{\alpha }_{ij}}}} \right\rangle ; & \text{for cost type attribute} \\ \left\langle {{\nu }_{{{\alpha }_{ij}}}},{{\mu }_{{{\alpha }_{ij}}}},{{\mu }_{{{\alpha }_{ij}}}} \right\rangle ; & \text{for benefit type attribute} \\\end{matrix} \right.$ (5)
Step 2: Calculate the FM on S_{j} using definition 5.
Step 3: Confirm the weighting vector w=(w_{1}, w_{2}, ..., w_{n})^{T} with respect to the orders 0≤w_{j}≤1 and $\sum_{i=1}^n w_j=1$.
Step 4: Use the CIPFHGA operator to combine the evaluation for each alternative under each criteria, such that $\alpha_i=C I P F H G A\left(\alpha_{i 1}, \alpha_{i 2}, \cdots, \alpha_{i n}\right)$.
Step 5: Select the best alternative by ranking the alternatives based on the score and accuracy values of α_{j}.
Figure 1. Flowchart for the selection of the best alternative
In this part, we will use the numerical example below to demonstrate the proposed approach. Assume that decision makers must select between four alternatives A_{1}, A_{2}, A_{3} and A_{4}. Four attributes S_{1}, S_{2}, S_{3} and S_{4} where the attributes are cost attributes. The PF ratings of alternatives {A_{i}}i=1, 2, ..., m according to attributes{S_{j}}j=1, 2, ..., n are evaluated by decision makers and form the decision matrix is given below and the weight information of attribute w=(0.1550, 0.3450, 0.3450, 0.1550)^{T} by referring to (Xu [26]).
$\begin{align} & \begin{matrix} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{1}} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{2}} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{3}} & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{A}_{4}} \\\end{matrix} \\ & \begin{matrix} {{S}_{1}} \\ {{S}_{2}} \\ {{S}_{3}} \\ {{S}_{4}} \\\end{matrix}\left[ \begin{matrix} \left\langle 0.5,0.2,0.3 \right\rangle & \left\langle 0.2,0.3,0.1 \right\rangle & \left\langle 0.2,0.3,0.3 \right\rangle & \left\langle 0.7,0.2,0.1 \right\rangle \\ \left\langle 0.4,0.3,0.2 \right\rangle & \left\langle 0.6,0.2,0.1 \right\rangle & \left\langle 0.4,0.3,0.3 \right\rangle & \left\langle 0.5,0.2,0.3 \right\rangle \\ \left\langle 0.6,0.1,0.1 \right\rangle & \left\langle 0.4,0.2,0.3 \right\rangle & \left\langle 0.5,0.2,0.3 \right\rangle & \left\langle 0.3,0.4,0.1 \right\rangle \\ \left\langle 0.5,0.3,0.1 \right\rangle & \left\langle 0.7,0.1,0.1 \right\rangle & \left\langle 0.4,0.3,0.2 \right\rangle & \left\langle 0.6,0.1,0.2 \right\rangle \\\end{matrix} \right] \\\end{align}$
Step 1: There is no need to convert the criteria because they all cost.
Step 2: Evaluate the FM on S_{j}. The FM are sets as follows:
$\vartheta \left( {{S}_{1}} \right)=0.23,\,\vartheta \left( {{S}_{2}} \right)=0.31,\,\vartheta \left( {{S}_{3}} \right)=0.18,$
$\vartheta \left( {{S}_{4}} \right)=0.24,\,\vartheta \left( {{S}_{1}},{{S}_{2}} \right)=0.5482,$
$\vartheta \left( {{S}_{1}},{{S}_{3}} \right)=0.4148,\,\vartheta \left( {{S}_{1}},{{S}_{4}} \right)=0.4764,$
$\vartheta \left( {{S}_{2}},{{S}_{3}} \right)=0.4964,\,\vartheta \left( {{S}_{2}},{{S}_{4}} \right)=0.5586,$
$\vartheta \left( {{S}_{3}},{{S}_{4}} \right)=0.4249,\,\vartheta \left( {{S}_{1}},{{S}_{2}},{{S}_{3}} \right)=0.7396,$
$\vartheta \left( {{S}_{1}},{{S}_{2}},{{S}_{4}} \right)=0.8033,\,\vartheta \left( {{S}_{1}},{{S}_{3}},{{S}_{4}} \right)=0.6663,$
$\vartheta \left( {{S}_{2}},{{S}_{3}},{{S}_{4}} \right)=0.7501,\,\vartheta \left( {{S}_{1}},{{S}_{2}},{{S}_{3}},{{S}_{4}} \right)=1.$
Step 3: Confirm the weighting vector w=(0.1550, 0.3450, 0.3450, 0.1550)^{T} by referring to (Xu [26]).
Step 4: Calculate the evaluations. As an example, consider the calculation of A_{1}.
Since ${{\left\langle 0.5,0.2,0.3 \right\rangle }^{0.2468}}=\left\langle 0.2435,0.6722,0.0843 \right\rangle ,$ ${{\left\langle 0.2,0.3,0.1 \right\rangle }^{0.3114}}=\left\langle 0.1185,0.6873,0.0823 \right\rangle ,$${{\left\langle 0.2,0.3,0.3 \right\rangle }^{0.1938}}=\left\langle 0.0824,0.7919,0.0663 \right\rangle $ and ${{\left\langle 0.7,0.2,0.1 \right\rangle }^{0.2480}}=\left\langle 0.3033,0.6709,0.0258 \right\rangle$
Their respective score values are 0.523, 0.6511, 0.7763, and 0.3934. We get their order as:
$\begin{align} & \left\langle 0.3033,0.6709,0.0258 \right\rangle >\left\langle 0.2435,0.6722,0.0843 \right\rangle > \\ & \,\,\,\,\,\left\langle 0.1185,0.6873,0.0823 \right\rangle >\left\langle 0.0824,0.7919,0.0668 \right\rangle . \\\end{align}$
Hence, we have:
$\begin{align} & CIPFHGA\left( {{\alpha }_{11}},{{\alpha }_{12}},{{\alpha }_{13}},{{\alpha }_{14}} \right)= \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\left\langle 0.3033,0.6709,0.0258 \right\rangle }^{4\cdot 0.1550}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\otimes {{\left\langle 0.2435,0.6722,0.0843 \right\rangle }^{4\cdot 0.3450}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\otimes {{\left\langle 0.1185,0.6873,0.0823 \right\rangle }^{4\cdot 0.3450}} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\otimes {{\left\langle 0.0824,0.7919,0.0668 \right\rangle }^{4\cdot 0.1550}} \\\end{align}$
$\left\langle \begin{array}{c}(0.3033+0.6709)^{0.62}0.6709^{0.62} \\ \cdot(0.2435+0.6722)^{1.38}0.6722^{1.38}\\ \cdot(0.1185+0.6873)^{1.38}0.6722^{1.38} \\ \cdot(0.0824+0.7919)^{0.62}0.7919^{0.62},0.6709^{0.62} \\ \cdot 0.6722^{1.38} \cdot 0.6722^{1.38} \cdot 0.7919^{0.62}, \\ 1\left[(10.0258)^{0.62} \cdot(10.0843)^{1.38} \cdot(10.0823)^{1.38} \cdot(10.0668)^{0.62}\right] \end{array} \right\rangle$
$=\langle 0.0134,0.0252,0.6839\rangle$
At the same reason, A_{2 }= ⟨0.0016, 0.2426, 0.2300⟩, A_{3 }= ⟨0.0015, 0.1988, 0.2449⟩ and A_{4}=⟨0.0033, 0.1831, 0.1473⟩.
Step 5: Rank the alternatives. The A_{i}(i =1, 2, 3, 4) score values are 0.6957, 0.4710, 0.4422 and 0.3271 respectively. Hence alternative can be ordered as A_{4}>A_{3}>A_{2}>A_{1} the best alternative A_{4}.
We evaluated the proposed MADM problem to the existing AOs to demonstrate the utility and effectiveness of the obtained results.
6.1 Comparison with existing approaches
The following Table 3 compares the proposed CIPFHGA operator with several existing aggregation operators.
Table 3. A comparison of various existing approaches and the proposed approach
Authors 
Aggregate Operators 
Ranking Order 
Wei [8] 
PFWG 
$A_4>A_2>A_3>A_1$ 
Wang et al. [9] 
PFWG 
$A_4>A_2>A_3>A_1$ 
Garg [10] 
PFWA 
$A_4>A_2>A_3>A_1$ 
Wei [11] 
PFHWA 
$A_4>A_2>A_3>A_1$ 
Jana et al. [16] 
PFHWG 
$A_4>A_2>A_3>A_1$ 
Khan et al. [17] 
PFEWA 
$A_4>A_2>A_3>A_1$ 
Proposed operator 
CIPFHGA 
$A_4>A_3>A_2>A_1$ 
We compare the proposed CIPFHGA operator with six existing AOs [811, 16, 17] in Table 3. It can be noted that all the existing AOs obtain the same ranking results and the same optimal alternative, while the proposed operator also optimal alternative is same, which is A_{4} but obtain different ranking results from the existing AOs.
The suggested approach select option A_{4} as the best choice, which is consistent with the findings produced by previous AOs. This validates the suggested operators practicality. Combining the Choquet integral with the OWG operator, the CIPFHGA operator ensures that all criteria establish correlative relationships and position weights are taken into consideration, providing a robust solution for dealing with MADM problems.
6.2 Advantages of the proposed operators
The benefits of our proposed approach can be described based on the aforementioned comparisons [811, 16, 17].
A PFS is a more effective way of dealing with uncertainty in given information and it has a various of applications in DM. AOs are especially important in the DM process since they combine all of the provided individual assessment values into a single form. The correlative relationships between criteria sets in fuzzy information systems are well defined by FMs, which may be further dealt with using the Choquet integral. We introduced the CIPFGA operator as an aggregation operator inspired by the Choquet integral and also proposed the CIPFHGA operator further by combining weights position. Some main properties of these proposed operators have been demonstrated and the functioning of the operators have been validated with illustrations. Further, the proposed operators enable us to solve a numerical example in MADM problem with PF environments. Finally, we have compared the proposed operators with some existing AOs. The viability of the proposed technique is demonstrated by different results of the ranking alternatives.
The specified AOs can be enhanced in future to handle more complex environments, such as interval valued PF environments, triangular PF environments and trapezoidal PF environments. In addition, we will continue to work on extending and applying the proposed operators to other domains.
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